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Three Primes

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January 2, 2005
geometry3D geometrynumber theoryprime numbers

Problem Statement

Let's write p,q, and r as three distinct prime numbers, where 1 is not a prime. Which of the following is the smallest positive perfect cube leaving n \equal{} pq^2r^4 as a divisor? <spanclass=′latex−bold′>(A)</span> p8q8r8<spanclass=′latex−bold′>(B)</span> (pq2r2)3<spanclass=′latex−bold′>(C)</span> (p2q2r2)3<spanclass=′latex−bold′>(D)</span> (pqr2)3<spanclass=′latex−bold′>(E)</span> 4p3q3r3 <span class='latex-bold'>(A)</span>\ p^8q^8r^8\qquad <span class='latex-bold'>(B)</span>\ (pq^2r^2)^3\qquad <span class='latex-bold'>(C)</span>\ (p^2q^2r^2)^3\qquad <span class='latex-bold'>(D)</span>\ (pqr^2)^3\qquad <span class='latex-bold'>(E)</span>\ 4p^3q^3r^3
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